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A set of 15 different integers has median of 25 and a range 19 Jan 2018, 21:44
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So does that mean that this set does not contain any -ve integers? I am assuming it can not but why?
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The set cannot have negative integers because of the given data points.
The median is 25 so the middle element is 25.
The range is 25 so greatest - smallest = 25
We know that 25 is there in the set so the smallest element can certainly not be less than 0 since the range of the entire set is 25.
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A set of 15 different integers has median of 25 and a range 24 Nov 2018, 15:56
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A video explanation can be found here:
https://www.youtube.com/watch?v=z_TK8hXmny0&t=35s

Median is 25. In a set of 15 integers, the median will be the 8th number (take 15 – 1, then divide that by 2. You have seven numbers to left of the median and seven to the right.)

Maximum value of the “greatest” possible integer would equal maximum value of the “least” possible integer, plus 25.

Since all integers are different, least possible integer is 15 - 7 = 18.

Maximum possible integer is therefore 18 + 25 = 43
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A set of 15 different integers has median of 25 and a range 25 Dec 2018, 11:28
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A set of 15 different integers has median of 25 and a range of 25. What is greatest possible integer that could be in this set?

A. 32
B. 37
C. 40
D. 43
E. 50
Show more
WLOG (without loss of generality) we may assume that:

\(a = {x_1} < {x_2} < \ldots < {x_7} < {x_8} = 25 < {x_9} < \ldots < {x_{14}} < {x_{15}} = a + 25\,\,\,\,\,{\text{ints}}\)

Considering this powerful structure, the problem is trivialized:

\(?\,\, = \,\,\left( {a + 25} \right)\,\,\max \,\,\,\,\,\,\, \Leftrightarrow \,\,\,\,\,\,a\,\,\max\)

\(a\,\,\max \,\,\,\,\, \Leftrightarrow \,\,\,\,\left( {{x_7},{x_6},{x_5},{x_4},{x_3},{x_2},{x_1} = a} \right) = \left( {24,23,22,21,20,19,18} \right)\,\,\,\,\,\,\,\, \Rightarrow \,\,\,\,\,a\,\,\max \,\, = \,\,18\,\,\,\,\,\,\,\, \Rightarrow \,\,\,\,\,\,\,\,? = 43\)


This solution follows the notations and rationale taught in the GMATH method.

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Fabio.
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A set of 15 different integers has median of 25 and a range 21 Apr 2020, 01:33
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may i know is this a vice-versa rule i.e , when asked to find the greatest possible value of the smallest integer in the set, do we need to maximize the greater value of the set ?


Thanks in advance.


EMPOWERgmatRichC
Hi Pratyaksh2791,

This question asks us to find the greatest possible number that could be in this set. Since 25 is the MEDIAN of the group of 15 INTEGERS, we know that 7 integers are greater than 25 and 7 integers are less than 25. We're told that the largest integer is exactly 25 more than the smallest integer, so to maximize the biggest value, we also have to maximize the smallest value. Since we're restricted to INTEGERS, the only way to get that maximum result is if the 7 integers that are less than 25 are CONSECUTIVE integers:

24, 23, 22, 21, 20, 19 and 18

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A set of 15 different integers has median of 25 and a range 21 Apr 2020, 12:27
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SuryaNouliGMAT
may i know is this a vice-versa rule i.e , when asked to find the greatest possible value of the smallest integer in the set, do we need to maximize the greater value of the set ?


Thanks in advance.


EMPOWERgmatRichC
Hi Pratyaksh2791,

This question asks us to find the greatest possible number that could be in this set. Since 25 is the MEDIAN of the group of 15 INTEGERS, we know that 7 integers are greater than 25 and 7 integers are less than 25. We're told that the largest integer is exactly 25 more than the smallest integer, so to maximize the biggest value, we also have to maximize the smallest value. Since we're restricted to INTEGERS, the only way to get that maximum result is if the 7 integers that are less than 25 are CONSECUTIVE integers:

24, 23, 22, 21, 20, 19 and 18

GMAT assassins aren't born, they're made,
Rich
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Hi SuryaNouliGMAT,

When a GMAT question asks you to consider a 'set' of numbers, you have to pay careful attention to the information that you're given about the set. In this prompt, we're told that the RANGE = 25 which means something really specific (re: the largest number is "25 more" than the smallest number - so if you change one of those two numbers, then you have to change the other as well). With this question, to maximize one number, you also have to maximize the other.

In other situations, to make the smallest number as big as possible, you have to make the other numbers as SMALL as possible. For example:

"The sum of 5 distinct integers is 100. What is the maximum possible value of the smallest number in this group?"

Here, we know that the 5 integers are DIFFERENT and that the sum of the integers is 100. To maximize the smallest value, we have to make the other 4 integers as small as possible (while still making sure that they are each bigger than the smallest integer). That group would be 18, 19, 20, 21 and 22.

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A set of 15 different integers has median of 25 and a range 13 Oct 2020, 06:04
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Why can't lowest integer be less than 18 .Why it can not be let us say -18 ? Integers are different right .
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A set of 15 different integers has median of 25 and a range 13 Oct 2020, 15:40
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Hi Bhanu1134527,

This prompt tells us 3 specific facts about a group of numbers:
1) There are 15 DIFFERENT INTEGERS
2) The group has MEDIAN of 25
3) The group has a RANGE of 25.

If the lowest number was -18, then the largest number would be 7 (because the RANGE = 25).... but that does not match the 2nd piece of information that we're given (that the MEDIAN = 25). Thus, the smallest number CAN'T be -18.

We're asked to find the largest possible integer that could be in this set (based on the facts that we were given), so because of the 3rd rule (the RANGE = 25), we also have to make the smallest integer as big as possible.

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A set of 15 different integers has median of 25 and a range 23 Mar 2021, 03:53
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Similar from EXPERTS' GLOBAL: https://gmatclub.com/forum/a-set-of-31- ... l#p2748611
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A set of 15 different integers has median of 25 and a range 24 Mar 2021, 04:07
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bhushangiri
A set of 15 different integers has median 25 and range 25. What could be the greatest possible integer in this set ?

OA is 43.

How 43 ? I am getting 50.


Try to right down 25 in the middle as a median and 7 numbers to the left and 7 nubers to the right.
You will see clearly that the minimum possible least number is 18 to the left of 25.
Hence 18+25 -->43

18 19 20 21 22 23 24 25 ....... --> the least possible
Show more




Hii,
even I got 43 by just substituting options, however, i'm confused why are consecutive integers only taken into consideration here? it could also be spread out differently without leaving a common difference. right?
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A set of 15 different integers has median of 25 and a range 24 Mar 2021, 14:57
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Hi Jaya6,

Since the question asks for the LARGEST POSSIBLE integer that could be in the set of numbers, we have to tailor our work around a certain 'math idea' - since we have a range of 25, to get the largest possible integer, we need the smallest integer in the set to be as big as possible. Working 'downwards' from the median - and keeping in mind that all of the values are distinct - the numbers BELOW the median would have to be consecutive. In that way, we could make the smallest number as big as possible.

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A set of 15 different integers has median of 25 and a range 09 Sep 2022, 08:00
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EMPOWERgmatRichC
Hi All,

When it comes to maximizing or minimizing a value in a group of numbers, you have to think about what the other numbers would need to be to accomplish your goal.

Here, we have a group of 15 DISTINCT (meaning DIFFERENT) integers with a median of 25 and a RANGE of 25. That range will dictate how large the largest value can be.

With a median of 25, we know that 7 numbers are LESS than 25 and 7 numbers are GREATER than 25:

_ _ _ _ _ _ _ 25 _ _ _ _ _ _ _

To maximize the largest value, we need to maximize the smallest value. Here's how we can do it:

18 19 20 21 22 23 24 25 _ _ _ _ _ _ _

With 18 as the smallest value, and a range of 25, the largest value would be 43.

Final Answer:
Show Spoiler
D

GMAT assassins aren't born, they're made,
Rich
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I'm not clear on this statement "To maximize the largest value, we need to maximize the smallest value". How do you come to this inference?
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A set of 15 different integers has median of 25 and a range 09 Sep 2022, 14:33
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EMPOWERgmatRichC
Hi All,

When it comes to maximizing or minimizing a value in a group of numbers, you have to think about what the other numbers would need to be to accomplish your goal.

Here, we have a group of 15 DISTINCT (meaning DIFFERENT) integers with a median of 25 and a RANGE of 25. That range will dictate how large the largest value can be.

With a median of 25, we know that 7 numbers are LESS than 25 and 7 numbers are GREATER than 25:

_ _ _ _ _ _ _ 25 _ _ _ _ _ _ _

To maximize the largest value, we need to maximize the smallest value. Here's how we can do it:

18 19 20 21 22 23 24 25 _ _ _ _ _ _ _

With 18 as the smallest value, and a range of 25, the largest value would be 43.

Final Answer:
Show Spoiler
D

GMAT assassins aren't born, they're made,
Rich


I'm not clear on this statement "To maximize the largest value, we need to maximize the smallest value". How do you come to this inference?
Show more

Hi NikTek,

The prompt tells us that the RANGE of the numbers is 25, meaning that the difference between the largest number in the group and the smallest number in the group is 25. Since these two values differ exactly by 25, when one increases, the other increases - so to find the largest possible value, we need the smallest possible value to be as big as possible.

GMAT assassins aren't born, they're made,
Rich

Contact Rich at: [email protected]
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A set of 15 different integers has median of 25 and a range 13 Jan 2025, 18:22
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Why are we considering the max value for all the other numbers? If we have to find the greatest number, which is supposed to be the last term, shouldn't we assign minimum values to the rest?

I am very confused here. Please help

bhushangiri
A set of 15 different integers has median of 25 and a range of 25. What is greatest possible integer that could be in this set?

A. 32
B. 37
C. 40
D. 43
E. 50
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A set of 15 different integers has median of 25 and a range 13 Jan 2025, 23:00
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Bhuvan1212
Why are we considering the max value for all the other numbers? If we have to find the greatest number, which is supposed to be the last term, shouldn't we assign minimum values to the rest?

I am very confused here. Please help

bhushangiri
A set of 15 different integers has median of 25 and a range of 25. What is greatest possible integer that could be in this set?

A. 32
B. 37
C. 40
D. 43
E. 50
Show more

That would apply if the sum of the terms were fixed, which isn’t the case for this question. There’s an extensive two-page discussion on this topic, with many experts providing their solutions. Please review it carefully. Hope this helps.
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A set of 15 different integers has median of 25 and a range 16 Mar 2025, 17:20
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Range = Max - Min value

Here, Max - Min = 25 => Max = 25 + Min. value

This means, to find the max value, you need the highest min. value.

Let min. value = x
Since the set consists of 15 "different" integers and median of the set is 25, it means the x+7 = 25 => x = 18

Max value = 25 + 18 = 43
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A set of 15 different integers has median of 25 and a range 20 Aug 2025, 11:53
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In order for 25 to be the median result of a 15 integer set, we would need 7 integers less than 25 and 7 greater. This is true because each number in the set is different and an integer, plus we have an odd number of terms. In order to have 25 as the median we need to have 7 smaller numbers and if the range is 25 that means x-25=18 is the largest possible result.

D is correct
bhushangiri
A set of 15 different integers has median of 25 and a range of 25. What is greatest possible integer that could be in this set?

A. 32
B. 37
C. 40
D. 43
E. 50
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A set of 15 different integers has median of 25 and a range 01 Mar 2026, 22:41
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The key word of this problem is DIFFERENT. When I first did this problem, I thought the answer was 50. However, when you notice that it says "different integers" thats when you know the answer cannot be 50. Because they have to be different numbers, and 25 is the middle number, the very lowest number could be 18. Thus, add a range of 25 to that and we get our highest possible number which is 43 (18 + 25 = 43).
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A set of 15 different integers has median of 25 and a range 11 Mar 2026, 03:29
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Given:
  • A set consists of 15 different integers.
  • The set has a median of 25, and a range of 25.

To find: The greatest integer that could be in this set.

Solution:
MEDIAN:
  • The median of 15 terms is the 8th term upon arranging all the terms in ascending order.
  • So, the 8th term = 25.
  • Thus, there are 7 integers less than 25 in this set.

RANGE:
  • Now, since Range = Greatest – Least number, we have: If least number = L, then greatest number = L + 25.
  • We want to maximize (L + 25). This will be maximum when L is maximum.
  • Now, L, being the least, is definitely less than the median 25. (Since all numbers are different, we cannot have them all be 25.)
  • So, the maximum L can be is 25 – 7 = 18. (Reducing 25 by just 1 each time to get the term before it.)
  • Finally, since L(max) = 18, the maximum possible value of L + 25 is 18 + 25 = 43.

Correct Answer: D

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